Truss manipulators are widely used in industrial automation, capable of automatically executing pre-programmed trajectories and actions to achieve operations such as grabbing and transporting objects. The key components include the actuator, drive system, control software, and sensing equipment, which together ensure the efficient material handling of the robotic arm in industrial production, as shown in Figure 1.

As shown in Figure 1, the structure of the truss manipulator consists of a vacuum pump, a walking rack, an accumulator, a lifting beam, a lifting servo motor, a moving plate, a descent detection, and a suction cup frame. The vacuum pump is responsible for providing suction power, the walking rack guides the horizontal movement of the robotic arm, the accumulator maintains stable system pressure, and the lifting beam and lifting servo motor work together to achieve vertical movement of objects. The moving plate is used to carry and transport objects, the descent detection sensor monitors the descent process of objects, and the suction cup holder fixes the suction cup to grasp objects. Figure 2 shows the workflow of the truss manipulator.

As shown in Figure 2, as the control center of the truss manipulator, the industrial controller is responsible for executing logical control tasks. It analyzes input signals, makes decisions based on the analysis results, and commands execution components. These components operate synchronously according to the instructions of the controller, achieving multi-axis coordinated actions of the robotic arm to complete automation tasks. Automated robotic arms are susceptible to environmental and equipment fluctuations during operation, resulting in workpiece positioning errors that can reduce efficiency and even cause machine tool collisions. Therefore, the machine tool system requires strict error control and coordination with other systems to adapt to complex production requirements. In the motion of the truss manipulator, significant positioning errors can occur, compromising precision and operational safety. These discrepancies primarily stem from geometric inaccuracies inherent in the mechanical system. Key contributing factors include manufacturing and assembly tolerances, rotational deviations in the gripper, wear and tear on components like gears and racks, and thermal expansion of the guide rails due to temperature changes. Figure 3 provides a schematic representation of these primary geometric errors within the manipulator’s coordinate system, illustrating how they contribute to the overall positioning deviation.

In the motion of the truss manipulator, significant positioning errors can occur due to factors such as manufacturing, assembly, gripper rotation, gear and rack wear, and changes in guide rail temperature (Yaldız et al., 2016). To improve the positioning accuracy, an NN trained model combined with PID control algorithm is used for error compensation.

As shown in Figure 4, in the research of optimizing the motion path of the truss manipulator, it is necessary to first select a suitable factor encoding method to represent the motion path parameters of the manipulator. Subsequently, a path optimization objective function is constructed, which is used to evaluate the optimization process and guide the search direction. It adjusts weights through BP NN to further refine the motion path. Meanwhile, a comprehensive explanation of the weight optimization matter in the BP algorithm is provided to give a theoretical basis for the algorithm. During the SA process, the energy differences of different paths are calculated and annealing operations are performed to explore the solution space. The fitness of each solution is evaluated to determine its optimization potential. Ultimately, the initial solution is crafted to serve as the algorithm’s starting point, from which it iteratively seeks out the optimal path.

In this study, path smoothness is a critical metric used to evaluate the quality of the manipulator’s trajectory. It is quantified as the average cosine of the angle between consecutive path segments. For a path defined by a sequence of points P 1 , P 2 , … , P N , the smoothness S is calculated as shown in Formula 1. S = 1 N − 2 ∑ i = 2 N − 1 P i − P i − 1 · P i + 1 − P i ∥ P i − P i − 1 ∥ ∥ P i + 1 − P i ∥ (1)

This metric yields a value between −1 and 1, where a value closer to one indicates a smoother path with minimal changes in direction. Path smoothness is highly relevant as it directly correlates with real-world performance. A smoother path reduces mechanical stress and vibrations on the manipulator’s joints and structure, leading to less wear and tear. Furthermore, it improves energy efficiency by minimizing abrupt accelerations and decelerations, and enhances stability when handling delicate or liquid materials (Kara et al., 2015).

To optimize the motion path of the truss manipulator, research needs to be conducted according to the process shown in Figure 4. SAA is used for path planning of truss manipulator, mimicking the annealing process of material heating and gradually cooling. As an extension of the random search algorithm, it probabilistically accepts changes in the solution and adjusts the grope method over time to locate the overall best solution, decreasing the danger of falling into LO. The SAA model is shown in Figure 5.

Figure 5 illustrates the core mechanism of the SAA, which is designed to find a GO solution by mimicking the physical process of annealing. The algorithm begins at an initial solution (State A) and explores the solution space by generating neighboring solutions (e.g., State B). The decision to move to a new state is governed by two principles. First, if the new solution is an improvement (i.e., has lower ‘energy’), it is always accepted, guiding the search towards better solutions. Second, and more critically, if the new solution is worse (higher ‘energy’), it may still be accepted based on a certain probability. This probabilistic acceptance allows the algorithm to make temporary ‘uphill’ moves, enabling it to escape LO that would trap simpler algorithms. As the algorithm progresses, a ‘temperature’ parameter is gradually lowered, which systematically reduces the probability of accepting worse solutions, ensuring that the search eventually converges towards the globally optimal state. The SAA combined with BP NN is represented in Figure 6.

As represented in Figure 6, in the research on optimizing the motion path of the truss manipulator, SAA and BP NN are used to train the critical path parameters. Through a three-layer NN model, SAA is used to initialize network weights and avoid LO, while BP NN adjusts weights to optimize paths. The objective function evaluates the path efficiency to ensure that the robotic arm plans its motion path quickly and efficiently, improves production efficiency, and ensures operational safety (Agrawal et al., 2024). In this training model, the amount of input points is n , and the input layer (IL) points of the network correspond to the teaching quality evaluation indicators, as shown in Formula 2. p i = 1 , 2 , ⋯ , n (2)

In Formula 2, p i means the output signal of the i th input node, and the number of motion parameters of the robotic arm corresponds to the amount of nodes in the IL of the NN. These parameters include position, velocity, and acceleration, which serve as input signals to the intermediate layer of the network after processing. Therefore, for the IL of an NN, the received motion parameters are consistent in quantity with the information output to the intermediate layer. The number of intermediate layer points is n , and the calculation for the network layer input is represented in Formula 3. I j = ∑ i = 1 n ω i j p i (3)

In Formula 3, I j means the input signal of the j th neuron, and ω i j means the contribution of the i th input node to the input signal of the j th neuron. The calculation for the output of the intermediate layer is shown in Formula 4. O j = 1 / 1 + ∑ i = 0 n − 1 ω i j p i − 1 − 1 2 = 1 / 1 + I j − 1 − 1 2 (4)

In Formula 4, O j represents the output of the j th intermediate node. The final output layer (OL) node number is 1. To convert a real value into a value between 0 and 1, the calculation is shown in Formula 5. q = 1 1 + ∑ i = 0 n − 1 w j O j − 1 − 1 2 (5)

In Formula 5, q represents the probability of an event occurring. O j means the value of the j th input feature. w j means the contribution of input features to the output variable. In the SA-BP algorithm, for the purpose of facilitating genetic operations, the SAA chooses binary encoding to genetically encode the weight coefficients. The training objective function is determined, whose mathematical expression is shown in Formula 6. E = 1 / p ∑ p = 1 P q ¯ − q 2 = 1 / p ∑ p = 1 P E j (6)

In Formula 6, E represents the mean square error, which is the average of the squares of all individual errors. q represents the actual value of a single observation point. q ¯ represents the average of actual values. p means the amount of samples. E j means the square of the error at the j th observation point. In the process of adjusting the weight coefficients, the gradient descent method is used, and the adjustment formula for ω i j is shown in formula 7. Δ ω i j = − η ∂ E / ∂ ω i j (7)

In Formula 7, Δ ω i j represents the change in weight; η represents the learning rate; ∂ E / ∂ ω i j means the partial derivative of the loss function E with respect to the weight ω i j . The adjustment amount of w j is shown in Formula 8. Δ w j = − η ∂ E / ∂ w j (8)

In Formula 8, Δ w j represents the change in w j . The weight Δ ω i j is further updated as shown in Formula 9. Δ ω i j = x i w j O j 2 1 − ∑ i = 1 n ω i j x i δ j (9)

In Formula 9, x i represents the activation value or input signal of the i th input node. δ j means the error derivative of the j th neuron, used to propagate errors and update weights, as shown in Formula 10. δ j = 2 q 2 1 − ∑ j = 1 j w j O j q ¯ − q 2 (10)

Formula 10 is used to update the weights in the network to reduce the discrepancy between the forecasted result and the real target value. When studying the optimization of the motion path of the truss manipulator, the MPU6050 sensor is used to obtain real-time posture data of the manipulator, and the BP NN is applied to forecast the optimal motion trajectory. By combining SAA to optimize PID control parameters, the system can reduce actual motion deviation and ensure precise control. u k = K p e k + K ∑ e k + K D e k − e k − 1 (11)

In Formula 11, u k represents the control input or control action at discrete time step k . K p represents proportional gain. e k means the error at discrete time step k . K D represents differential gain. A BP NN with a single hidden layer was constructed in the study. The OL is designed with three output nodes, corresponding to the K p , K i , and K d parameters of the PID controller. Each node outputs a non-negative value, thus forming a three neuron OL. Since the input and output of the IL are the same, it can be expressed as Formula 12. o j 1 = x j (12)

In Formula 12, o j 1 means the output of the j th neuron in the first layer of the NN. x j represents the j th element in the input vector. The input and output of the hidden layer in the network are shown in Formula 13. n e t i 2 k = ∑ j = 0 m w i j 2 o j 1 (13)

In Formula 13, n e t i 2 k represents the weighted sum of the inputs of the i th neuron in the second layer of the NN at the discrete-time step k . w i j 2 represents the weight between the j th neuron in the first layer and the i th neuron in the second layer. m represents the number of neurons in the first layer. The output of the hidden layer neurons in the time step is shown in Formula 14. o i 2 k = f n e t i 2 k (14)

The recursive formula for updating the weights of the third layer is shown in Formula 15. Δ w l i 3 k = − η ∂ E k ∂ w l i 3 + γ Δ w l i 3 k − 1 (15)

In Formula 15, Δ w l i 3 k represents the update amount of the weight w l i of the third layer at time step k . w l i represents the learning rate. γ represents the momentum factor. The final update is shown in Formula 16. Δ w i j 2 k = η δ i 2 o j 1 k + γ Δ w l i 2 k − 1 δ i 2 = f ′ n e t i 2 k ∑ l = 1 3 δ l 3 w l i 3 k   i = 1 , 2 , 3 . . . (16)

In Formula 16, δ i 2 represents the error derivative of the i th neuron in the second layer of the BP NN assisted by SAA. This error derivative is the key to the calculation in the backpropagation process and directly affects the update of hidden layer weights, thereby affecting the optimization effect of the robotic arm motion path. By accurately calculating the error derivative of each neuron, the network weights can be effectively adjusted to make the motion path of the robotic arm closer to the ideal state.

The performance assessment metrics for the model were studied through three metrics: Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE). The SA-BP algorithm was used to verify the optimization of the motion path of the truss manipulator. The network structure adopted a three-layer NN, with 10 IL nodes corresponding to multiple motion parameters of the truss manipulator. The amount of points in the hidden layer varied between 10 and 30. The OL was a single node that output the optimized motion path. The learning rate ranged from 0.0004 to 0.0012, and the gradient descent method was applied to optimize the model throughout the training phase. To determine the optimal hyper-parameters for the SA-BP model, a systematic tuning strategy based on Grid Search combined with 5-fold Cross-Validation was implemented. This approach ensured that the selected parameters were robust and not overfitted to a specific data split. The hyper-parameter space was defined based on preliminary experiments and common practices. Specifically, the number of hidden layer nodes was explored in the range of [10, 30] with an integer step of 2, while the learning rate was searched within the range of [0.0004, 0.0012] with a step of 0.0002. For each combination of hyper-parameters in the grid, the model was trained and evaluated using 5-fold cross-validation on the training dataset. The performance was measured by the average RMSE across the five validation folds. The combination yielding the lowest average RMSE was selected as the optimal configuration for the final model. The training data was the actual data of multiple motion paths of the robotic arm. This historical data were collected from a physical truss manipulator system as depicted in Figure 1. The data acquisition process involved executing a series of predefined motion tasks, such as pick-and-place operations, across various start and end points within the manipulator’s workspace. An MPU6050 inertial measurement unit was used to capture real-time posture data, including position, velocity, and acceleration for each axis. The dataset was structured as a time-series collection of data points, where each point contained a timestamp, the Cartesian coordinates (X, Y, Z) of the end-effector, its corresponding velocity and acceleration values, and the motor control signals sent from the controller. The error of SA-BP algorithm in optimizing the motion path of truss manipulator is shown in Figure 7.

Figure 7a shows the trend of three indicators as the amount of hidden layer nodes rose from 10 to 30. When the amount of hidden layer points was 20, the MAE reached the lowest value of 0.211, while the RMSE and MAPE were 0.282 and 0.347, separately, indicating that the model performed best at this number of nodes. Figure 7b shows the variation of model error as the learning rate was adjusted from 0.0004 to 0.0012. When the learning rate was 0.0008, the error values of MAE, RMSE, and MAPE were 0.942, 1.466, and 2.513, separately. At this time, the model’s error was relatively lowest, indicating that the model performed best at this learning rate. To enhance the algorithm performance, a three-layer SA-BP NN model was applied in the validation experiment. The IL had 10 nodes representing the key motion parameters of the truss manipulator, the hidden layer has 20 nodes, and the OL output the optimized path for a single node. The model used mean square error (MSE) as the loss function, with a learning rate of 0.0008, and was optimized through gradient descent. The model was trained and tested using historical motion data, and its stability was ensured through cross validation. The loss convergence curves of the SA-BP algorithm training and testing sets are shown in Figure 8.

On the training set shown in Figure 8a, the mean square error rapidly decreases in the initial stage, then continues to steadily decrease and converges to nearly 10⁻² around 1.3 × 10⁻⁴ iterations. Although there are slight fluctuations caused by random optimization, the overall trend is smooth and consistent; On the test set shown in Figure 8b, there is also monotonic convergence, with a rapid decline in the initial stage and a slow approach to the low error interval, reaching a low value of the same magnitude as the training set around 1.5-1.6 × 10⁻⁴ iterations. The continuous decline and convergence of the training and testing curves indicate that the model performs well in both optimization efficiency and generalization performance. It can quickly reduce errors without significant overfitting, thus demonstrating the advantages of the proposed method in terms of stability and prediction accuracy. To first evaluate the core discriminative power of the proposed model against other algorithms, a binary classification task was established as a preliminary assessment. In this framework, historical motion paths from the training dataset were labeled as either ‘acceptable’ (Class 1) or ‘unacceptable’ (Class 0) based on a predefined threshold of performance, such as exceeding a critical positioning error or failing to meet smoothness criteria. The performance of the SA-BP algorithm was benchmarked against four other common optimization algorithms (GA, PSO, DE, and ABC) on a binary classification task, with the results detailed in Table 1. This comparison aimed to assess the core discriminative power of each model in distinguishing between ‘acceptable’ and ‘unacceptable’ motion paths.

As shown in Table 1, the SA-BP algorithm demonstrated superior performance across all key metrics. It achieved the highest accuracy (93.78%), precision (91.62%), and recall (92.34%). Crucially, it also obtained the top F1 score of 91.98%, indicating the best balance between precision and recall, and the highest ROC-AUC value of 0.954, signifying its excellent overall capability to discriminate between the two classes. While other algorithms like ABC and PSO also yielded strong results, with F1 scores of 91.12% and 90.71% respectively, they did not surpass the balanced performance of the SA-BP model. The GA and DE algorithms showed comparatively weaker performance in this task. Overall, the comprehensive results in Table 1 confirmed that the SA-BP algorithm possessed the most robust and reliable discriminative power for evaluating the quality of manipulator motion paths.

The SA-BP algorithm combined with a PID controller was used in the experiment to optimize the motion path of the truss manipulator. The PID controller was used to adjust the motion trajectory of the robotic arm to minimize its error. The parameter settings were as follows: the initial value of proportional gain K _p was 0.1, the initial value of integral gain K _i was 0.05, and the initial value of differential gain K _d was 0.01. The sampling period of this experiment was 0.01 s, and the total running time was 3 s. The actual motion data of the robotic arm were fed back to the controller in real-time through sensors, and the PID parameters were optimized using BP NN and SAA to make the movement trajectory of the robotic arm smoother and more accurate. The time-varying curves of the three parameters of the PID controller are shown in Figure 9.

Figure 9 shows the changes in three parameters K _p , K _i , and K _d over time when the SA-BP algorithm was combined with a PID controller in the motion path optimization task of a truss manipulator. Firstly, there was a change in the K _p parameter, which rapidly increased in the initial stage and stabilized at around 0.15 at approximately 0.1 s. Next was the K _i parameter, which rapidly increased to around 0.11 at the beginning and reached a stable value at about 0.2 s. The final curve showed the variation of K _d parameter, which rapidly increased to 0.02 at about 0.05 s and remained unchanged. All three parameters converged to stable values within a brief timeframe, indicating that the algorithm had a fast and stable response speed in PID control. The comparison of optimization algorithm effectiveness for the motion path of the truss robotic arm is represented in Table 2.

As shown in Table 2, the SA-BP algorithm performed excellently in key performance indicators, with a path length of 12.486 m, the shortest, energy optimization reaching 23.78%, calculation time of 8.213 s, iteration convergence speed of 147 times, path error of 2.14 cm, smoothness of 0.854, demonstrating high stability. In contrast, the GA path length was 13.168 m, with an optimized energy consumption of 21.29% and a path error of 3.56 cm, showing slightly inferior performance. The PSO algorithm had a path length of 12.934 m, energy optimization of 22.46%, and 182 iterations, slightly slower than SA-BP. The DE algorithm had a path length of 13.513 m, a computation time of 9.315 s, 215 iterations, and the slowest convergence speed. The ABC algorithm had a path length of 12.763 m, optimized energy consumption of 22.35%, smoothness of 0.842, computation time of 10.817 s, balanced performance, and was close to SA-BP. Overall, the SA-BP algorithm demonstrated significant advantages in path optimization, energy consumption, and computational efficiency. The performance of the proposed SA-BP algorithm was comprehensively validated by comparing its generated path against both the original, unoptimized path and a theoretically optimal path, with the results detailed in Table 3.

As shown in Table 3, the algorithm demonstrated substantial improvements over the pre-optimization baseline across all evaluation metrics. Notably, it reduced the path length from 14.872 m to 12.486 m (a 16.05% improvement) and slashed the positioning error by a remarkable 62.09%, from 3.64 cm down to 1.38 cm. Crucially, the performance of the SA-BP optimized path closely approached the benchmarks set by the theoretically optimal path. The optimized path length, energy consumption, and execution time showed minimal deviation rates of just 1.16%, 1.18%, and 1.34% from the ideal values, respectively. Furthermore, key performance indicators such as smoothness (0.854 units) and stability (94.32%) were also remarkably close to the theoretical best-case scenario, with deviations of less than 1%. This proximity to the ideal benchmark strongly underscored the algorithm’s high efficiency and its capability to converge on a near-optimal solution. To rigorously validate the real-world applicability of the proposed SA-BP algorithm, a final verification experiment was conducted on the physical truss manipulator system described in Section 2.1. In this experiment, a standard pick-and-place task was defined with specified start and end coordinates. The SA-BP algorithm was first used to generate the optimal motion path and corresponding control commands. These commands were then fed directly to the manipulator’s control system. The actual executed trajectory of the manipulator’s end-effector was tracked in real-time using the MPU6050 sensor system. The performance of the predicted path was then compared against the actual executed path. Key performance indicators (KPIs) were measured to quantify the fidelity of the real-world execution against the algorithm’s optimized plan. The comparison between the SA-BP predicted path and the actual execution path on the physical system is shown in Table 4.

The results of this direct comparison are presented in Table 4, demonstrating a high degree of correlation between the predicted performance and the physically realized results, thereby confirming the algorithm’s effectiveness and robustness in a practical application scenario.